This paper concerns the solvability of a nonlinear fractional boundary value problem at resonance. By using fixed point theorems we prove that the perturbed problem has a solution, then by some ideas from analysis we show that the original problem is solvable. An example is given to illustrate the obatined results.

Boundary value problems (BVP) at resonance have been studied in many papers for ordinary differential equations (Feng and Webb 1997; Guezane-Lakoud and Frioui 2013; Guezane-Lakoud and Kılıçman 2014; Hu and Liu 2011; Jiang 2011; Kosmatov 2010, 2006; Mawhin 1972; Samko et al. 1993; Webb and Zima 2009; Zima and Drygas 2013), most of them considered the existence of solutions for the BVP at resonance making use of Mawhin coincidence degree theory (Liu and Zhao 2007). In Guezane-Lakoud and Kılıçman (2014), Han investigated the existence and multiplicity of positive solutions for the BVP at resonance by considering an equivalent non resonance perturbed problem with the same conditions. More precisely, he wrote the original problem \(u^{\prime \prime }=f\left( t,u\right)\) as

under the conditions \(\beta \in \left( 0,\frac{\pi }{2}\right)\) and \(f:[ 0,1] \times [ 0,\infty [ \rightarrow {\mathbb {R}}\) is continuous and \(f\left( t,u\right) \ge -\beta ^{2}u.\) This result has been improved by Webb et al., in Samko et al. (1993) where the authors investigated a similar problem with various nonlocal boundary conditions.

In a recent study Mawhin (1972), Nieto investigated a resonance BVP by an other approach, that we will apply to a fractional boundary value problem to prove the existence of solutions.

The goal of this paper is to provide sufficient conditions that ensure the existence of solutions for the following fractional boundary value problem (P)

has \(u(t)=ct^{2},\)\(c\in {\mathbb {R}}\) as nontrivial solutions. In this case since Leray-Schauder continuation theory cannot be used, we will apply some ideas from analysis. Although these techniques have already been considered in Mawhin (1972) for ordinary differential equation but the present problem (P) is different since the nonlinearity f depends also on the derivative and the differential Eq. (1) is of fractional type.

Fractional boundary value problems at resonance have been investigated in many works such in Bai (2011), Han (2007), Infante and Zima (2008), where the authors applied Mawhin coincidence degree theory. Further for the existence of unbounded positive solutions of a fractional boundary value problem on the half line, see Guezane-Lakoud and Kılıçman (2014).

The organization of this work is as follows. In Sect. 2, we introduce some notations, definitions and lemmas that will be used later. Section 3 treats the existence and uniqueness of solution for the perturbed problem by using respectively Schaefer fixed point theorem and Banach contraction principal. Then by some analysis ideas, we prove that the problem (P) is solvable. Finally, we illustrate the obtained results by an example.

The first condition in (3) gives \(c_{0}=c_{1}=0,\) the second one implies that \(I_{0^{+}}^{q}y(1)=0,\) hence (3) has solution if and only if \(I_{0^{+}}^{q}y(1)=0\), then the problem (3) has an infinity of solutions given by

remarking that H(t, s) is continuous according to both variables s, t on \(\left[ 0,1\right] \times \left[ 0,1\right]\), nonnegative and \(0\le H(t,s)\le 2\) then using assumptions (14) and (15), we get

where \(f\in C\left( \left[ 0,1\right] \times {\mathbb {R}} \times {\mathbb {R}} ,{\mathbb {R}} \right)\), \(2<q<3,\)\(^{c}D_{0^{+}}^{\alpha }\) denotes the Caputo’s fractional derivative. By using fixed point theorems we proved that the perturbed problem has a solution, then we also show that the original problem is solvable. An example is provided n order to illustrate the results.

Authors' contributions

All authors read and approved the final manuscript

Acknowledgements

First of all the authors would like to thank the referees for giving useful suggestions to improve the manuscript. The first author would also like to thank the University Putra Malaysia for the kind hospitality during her visit in December 2015. The third author acknowledges that this research was partially supported by the University Putra Malaysia.

Competing interests

The authors declare that they have no competing interests.

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